1. ## Functions

Hey guys im preparing for a math contest and would like to understand how to do these questions. if anyone has any ideas please let me know.
thanks.

2. Originally Posted by ballaholic8
Hey guys im preparing for a math contest and would like to understand how to do these questions. if anyone has any ideas please let me know.
thanks.
to #16.

There are 2 different ways:

1. Use implicite differentiation:

$x^2+y^2=25~\implies~2x+2y\cdot y'=0$

Plug in the coordinates of the tangent point and solve for y':

$8-6\cdot y'=0~\implies~y'=\dfrac43$

Therefore the equation of the tangent is

$y+3=\dfrac43(x-4)~\implies~\boxed{y=\dfrac43x-\dfrac{25}3}$

2. $x^2+y^2=25$ is the equation of a circle with the center at the origin and with a radius of 5. The direction from the center to the tangent point is:

$m=\dfrac{-3}4$ The direction of the tangent must be perpendicular to this direction and therefore

$m_t=-\dfrac1{-\frac34} = \dfrac43$

Use the point-slope-formula of a straight line to get the equation of the tangent.

3. any ideas to 15?